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ABSTRACT.A notion of generalized bounded variation is introduced which simultaneously generalizes many of those previously examined. It is shown that the class of functions arising from this definition is a Banach space with a suitable norm. Appropriate variation functions are defined and examined, and an analogue of Helly's theorem is estabished.The significance of this class to convergence of Fourier series is briefly discussed.A result concerning Riemann-Stieltjes integrals of functions of this class is proved. CHAPTER IWe consider real valued functions whose domains are a closed, bounded interval, say [a, 6], to be made precise only if necessary. In the first section of this chapter we define a new notion of generalized bounded variation and the appropriate interval and point variation functions.We note some properties of functions which fall in this class, and observe how this new class is related to those already known. In §2 we examine the variation functions further, and investigate aspects of the linear structure of this class, showing that it is a Banach space and obtaining an analogue of Helly's theorem. In §3 we extend a theorem of Waterman concerning the convergence of Fourier series.

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